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(* ========================================================================= *)
(* Some special/counterintuive/weird topological spaces for counterexamples. *)
(* ========================================================================= *)
needs "Multivariate/metric.ml";;
(* ------------------------------------------------------------------------- *)
(* The indiscrete (trivial) topology. *)
(* ------------------------------------------------------------------------- *)
let indiscrete_topology = new_definition
`indiscrete_topology (u:A->bool) = topology {{},u}`;;
let OPEN_IN_INDISCRETE_TOPOLOGY = prove
(`!u s:A->bool.
open_in (indiscrete_topology u) s <=> s = {} \/ s = u`,
REPEAT GEN_TAC THEN REWRITE_TAC[indiscrete_topology] THEN
W(MP_TAC o fst o EQ_IMP_RULE o PART_MATCH (lhand o rand)
(CONJUNCT2 topology_tybij) o rator o lhand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[istopology; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY; FORALL_SUBSET_INSERT;
UNIONS_INSERT; SUBSET_EMPTY; FORALL_UNWIND_THM2; UNIONS_0;
UNION_EMPTY; INTER_EMPTY; INTER_IDEMPOT] THEN
REWRITE_TAC[IN_INSERT];
DISCH_THEN SUBST1_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM IN] THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY]]);;
let TOPSPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. topspace(indiscrete_topology u) = u`,
REWRITE_TAC[topspace; OPEN_IN_INDISCRETE_TOPOLOGY] THEN SET_TAC[]);;
let CLOSED_IN_INDISCRETE_TOPOLOGY = prove
(`!u s:A->bool.
closed_in (indiscrete_topology u) s <=> s = {} \/ s = u`,
REWRITE_TAC[closed_in; OPEN_IN_INDISCRETE_TOPOLOGY;
TOPSPACE_INDISCRETE_TOPOLOGY] THEN
SET_TAC[]);;
let SUBTOPOLOGY_INDISCRETE_TOPOLOGY = prove
(`!u s:A->bool.
subtopology (indiscrete_topology u) s =
indiscrete_topology (u INTER s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[TOPOLOGY_EQ] THEN
GEN_TAC THEN REWRITE_TAC[OPEN_IN_SUBTOPOLOGY_ALT] THEN
REWRITE_TAC[OPEN_IN_INDISCRETE_TOPOLOGY] THEN
ONCE_REWRITE_TAC[SET_RULE `x = a \/ x = b <=> x IN {a,b}`] THEN
REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; INTER_EMPTY] THEN
REWRITE_TAC[INTER_COMM]);;
let COMPACT_SPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. compact_space (indiscrete_topology u)`,
GEN_TAC THEN REWRITE_TAC[COMPACT_SPACE; OPEN_IN_INDISCRETE_TOPOLOGY] THEN
X_GEN_TAC `U:(A->bool)->bool` THEN STRIP_TAC THEN
EXISTS_TAC `U:(A->bool)->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN
MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{{},(u:A->bool)}` THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN
GEN_REWRITE_TAC I [SUBSET] THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY]);;
let COMPACT_IN_INDISCRETE_TOPOLOGY = prove
(`!u s:A->bool.
compact_in (indiscrete_topology u) s <=> s SUBSET u`,
REWRITE_TAC[COMPACT_IN_SUBSPACE; TOPSPACE_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[SUBTOPOLOGY_INDISCRETE_TOPOLOGY;
COMPACT_SPACE_INDISCRETE_TOPOLOGY]);;
let LOCALLY_COMPACT_SPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. locally_compact_space (indiscrete_topology u)`,
SIMP_TAC[COMPACT_IMP_LOCALLY_COMPACT_SPACE;
COMPACT_SPACE_INDISCRETE_TOPOLOGY]);;
let COMPLETELY_REGULAR_SPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. completely_regular_space(indiscrete_topology u)`,
REWRITE_TAC[completely_regular_space; CLOSED_IN_INDISCRETE_TOPOLOGY;
TOPSPACE_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[SET_RULE
`(s = {} \/ s = u) /\ x IN u DIFF s <=> s = {} /\ x IN u`] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2] THEN
MAP_EVERY X_GEN_TAC [`u:A->bool`; `x:A`] THEN
DISCH_TAC THEN EXISTS_TAC `(\x. &0):A->real` THEN
REWRITE_TAC[CONTINUOUS_MAP_CONST; NOT_IN_EMPTY] THEN
REWRITE_TAC[TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY;
ENDS_IN_UNIT_REAL_INTERVAL]);;
let REGULAR_SPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. regular_space(indiscrete_topology u)`,
SIMP_TAC[COMPLETELY_REGULAR_IMP_REGULAR_SPACE;
COMPLETELY_REGULAR_SPACE_INDISCRETE_TOPOLOGY]);;
let T1_SPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. t1_space(indiscrete_topology u) <=> ?a. u SUBSET {a}`,
REWRITE_TAC[T1_SPACE_CLOSED_IN_SING; TOPSPACE_INDISCRETE_TOPOLOGY;
CLOSED_IN_INDISCRETE_TOPOLOGY] THEN
SET_TAC[]);;
let T0_SPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. t0_space(indiscrete_topology u) <=> ?a. u SUBSET {a}`,
GEN_TAC THEN REWRITE_TAC[t0_space; OPEN_IN_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[TOPSPACE_INDISCRETE_TOPOLOGY; RIGHT_OR_DISTRIB] THEN
SIMP_TAC[EXISTS_OR_THM; UNWIND_THM2; NOT_IN_EMPTY] THEN SET_TAC[]);;
let HAUSDORFF_SPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. hausdorff_space(indiscrete_topology u) <=> ?a. u SUBSET {a}`,
MESON_TAC[REGULAR_T1_EQ_HAUSDORFF_SPACE; REGULAR_SPACE_INDISCRETE_TOPOLOGY;
T1_SPACE_INDISCRETE_TOPOLOGY]);;
let NORMAL_SPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. normal_space(indiscrete_topology u)`,
GEN_TAC THEN REWRITE_TAC[normal_space; CLOSED_IN_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[OPEN_IN_INDISCRETE_TOPOLOGY; RIGHT_OR_DISTRIB] THEN
REWRITE_TAC[EXISTS_OR_THM; GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[UNWIND_THM2] THEN SET_TAC[]);;
let DERIVED_SET_OF_INDISCRETE_TOPOLOGY = prove
(`!u s:A->bool.
(indiscrete_topology u) derived_set_of s =
if ?a. u INTER s SUBSET {a} then if DISJOINT u s then {} else u DIFF s
else u`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[DERIVED_SET_OF_RESTRICT] THEN
ONCE_REWRITE_TAC[DISJOINT; SET_RULE `u DIFF s = u DIFF (u INTER s)`] THEN
MP_TAC(SET_RULE `(u INTER s:A->bool) SUBSET u`) THEN
REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_INDISCRETE_TOPOLOGY] THEN
SPEC_TAC(`u INTER s:A->bool`,`t:A->bool`) THEN X_GEN_TAC `s:A->bool` THEN
STRIP_TAC THEN ASM_CASES_TAC `s:A->bool = {}` THEN
ASM_REWRITE_TAC[DERIVED_SET_OF_EMPTY; INTER_EMPTY; EMPTY_SUBSET] THEN
REWRITE_TAC[derived_set_of; EXTENSION; IN_ELIM_THM; IN_INTER] THEN
REWRITE_TAC[OPEN_IN_INDISCRETE_TOPOLOGY; TOPSPACE_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[TAUT
`p /\ (q \/ r) ==> s <=> (q ==> p ==> s) /\ (r ==> p ==> s)`] THEN
SIMP_TAC[NOT_IN_EMPTY; FORALL_UNWIND_THM2] THEN
X_GEN_TAC `x:A` THEN COND_CASES_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
ASM_CASES_TAC `(x:A) IN u` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; IN_DIFF] THEN
ASM SET_TAC[]);;
let DERIVED_SET_OF_TOPSPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool.
(indiscrete_topology u) derived_set_of u =
if ?a. u SUBSET {a} then {} else u`,
GEN_TAC THEN REWRITE_TAC[DERIVED_SET_OF_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[GSYM DISJOINT_EMPTY_REFL; INTER_IDEMPOT] THEN
REWRITE_TAC[DIFF_EQ_EMPTY; COND_ID]);;
let PERFECT_SET_INDISCRETE_TOPOLOGY = prove
(`!u s:A->bool.
(indiscrete_topology u) derived_set_of s = s <=>
s = {} \/ s = u /\ ~(?a. u SUBSET {a})`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `s:A->bool = {}` THEN
ASM_REWRITE_TAC[DERIVED_SET_OF_EMPTY] THEN
REWRITE_TAC[DERIVED_SET_OF_INDISCRETE_TOPOLOGY] THEN
ASM_CASES_TAC `s:A->bool = u` THENL [ALL_TAC; ASM SET_TAC[]] THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN
ASM_REWRITE_TAC[GSYM DISJOINT_EMPTY_REFL; INTER_IDEMPOT] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
let CONTINUOUS_MAP_INTO_INDISCRETE_TOPOLOGY = prove
(`!(f:A->B) top u.
continuous_map (top,indiscrete_topology u) f <=>
IMAGE f (topspace top) SUBSET u`,
REPEAT GEN_TAC THEN
REWRITE_TAC[continuous_map; OPEN_IN_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN
SIMP_TAC[NOT_IN_EMPTY; EMPTY_GSPEC; OPEN_IN_EMPTY] THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_UNWIND_THM2] THEN
REWRITE_TAC[TOPSPACE_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[TAUT `(p /\ q <=> p) <=> p ==> q`] THEN
SIMP_TAC[TAUT `p /\ q <=> ~(p ==> ~q)`] THEN
REWRITE_TAC[IN_GSPEC; OPEN_IN_TOPSPACE]);;
let CONTINUOUS_MAP_FROM_INDISCRETE_TOPOLOGY = prove
(`!top u (f:A->B).
t1_space top
==> (continuous_map(indiscrete_topology u,top) f <=>
u = {} \/ ?a. a IN topspace top /\ !x. x IN u ==> f x = a)`,
REWRITE_TAC[T1_SPACE_CLOSED_IN_SING] THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC `u:A->bool = {}` THEN
ASM_REWRITE_TAC[CONTINUOUS_MAP_CLOSED_IN; TOPSPACE_INDISCRETE_TOPOLOGY;
CLOSED_IN_INDISCRETE_TOPOLOGY; NOT_IN_EMPTY; EMPTY_GSPEC] THEN
EQ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
STRIP_TAC THEN MATCH_MP_TAC(SET_RULE
`~(u = {}) /\
(!x. x IN u ==> f x IN v) /\
(!x y. x IN u /\ y IN u ==> f x = f y)
==> ?a. a IN v /\ (!x. x IN u ==> f x = a)`) THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `b:A`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(fun th ->
MP_TAC(SPEC `{(f:A->B) b}` th) THEN MP_TAC(SPEC `{(f:A->B) a}` th)) THEN
ASM_SIMP_TAC[] THEN ASM SET_TAC[]);;
let PATH_CONNECTED_SPACE_INDISCRETE_TOPOLOGY = prove
(`!u:A->bool. path_connected_space(indiscrete_topology u)`,
REPEAT GEN_TAC THEN REWRITE_TAC[path_connected_space] THEN
REWRITE_TAC[path_in; CONTINUOUS_MAP_INTO_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[TOPSPACE_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[TOPSPACE_EUCLIDEANREAL_SUBTOPOLOGY] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN STRIP_TAC THEN
EXISTS_TAC `(\i. if i = &0 then x else y):real->A` THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM SET_TAC[]);;
let PATH_CONNECTED_IN_INDISCRETE_TOPOLOGY = prove
(`!u s:A->bool.
path_connected_in (indiscrete_topology u) s <=> s SUBSET u`,
REWRITE_TAC[path_connected_in; PATH_CONNECTED_SPACE_INDISCRETE_TOPOLOGY;
SUBTOPOLOGY_INDISCRETE_TOPOLOGY; TOPSPACE_INDISCRETE_TOPOLOGY]);;
let NEIGHBOURHOOD_BASE_OF_INDISCRETE_TOPOLOGY = prove
(`!P u:A->bool.
neighbourhood_base_of P (indiscrete_topology u) <=>
u = {} \/ P u`,
REWRITE_TAC[NEIGHBOURHOOD_BASE_OF; OPEN_IN_INDISCRETE_TOPOLOGY] THEN
REWRITE_TAC[TAUT
`(p \/ q) /\ r ==> s <=> (p ==> r ==> s) /\ (q ==> r ==> s)`] THEN
SIMP_TAC[IN_ELIM_THM; RIGHT_FORALL_IMP_THM; NOT_IN_EMPTY] THEN
REWRITE_TAC[FORALL_UNWIND_THM2; RIGHT_OR_DISTRIB] THEN
REWRITE_TAC[EXISTS_OR_THM; RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
REWRITE_TAC[NOT_IN_EMPTY; SUBSET_ANTISYM_EQ] THEN
ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> r /\ p /\ q`] THEN
SIMP_TAC[UNWIND_THM1] THEN SET_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Finite-complement (cofinite) topology. *)
(* ------------------------------------------------------------------------- *)
let cofinite_topology = new_definition
`cofinite_topology (u:A->bool) =
topology ({} INSERT {s | s SUBSET u /\ FINITE(u DIFF s)})`;;
let OPEN_IN_COFINITE_TOPOLOGY = prove
(`!u s:A->bool.
open_in (cofinite_topology u) s <=>
s = {} \/ s SUBSET u /\ FINITE(u DIFF s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[cofinite_topology] THEN
W(MP_TAC o fst o EQ_IMP_RULE o PART_MATCH (lhand o rand)
(CONJUNCT2 topology_tybij) o rator o lhand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[istopology];
DISCH_THEN SUBST1_TAC THEN
GEN_REWRITE_TAC LAND_CONV [GSYM IN] THEN
REWRITE_TAC[IN_ELIM_THM; IN_INSERT]] THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_INSERT] THEN
REWRITE_TAC[IN_INSERT; INTER_EMPTY] THEN CONJ_TAC THENL
[REWRITE_TAC[IN_ELIM_THM] THEN
X_GEN_TAC `s:A->bool` THEN STRIP_TAC THEN
X_GEN_TAC `t:A->bool` THEN STRIP_TAC THEN DISJ2_TAC THEN
REWRITE_TAC[SET_RULE `u DIFF s INTER t = (u DIFF s) UNION (u DIFF t)`] THEN
ASM_SIMP_TAC[FINITE_UNION] THEN ASM SET_TAC[];
X_GEN_TAC `k:(A->bool)->bool` THEN REWRITE_TAC[EMPTY_UNIONS] THEN
REWRITE_TAC[TAUT `p ==> q \/ r <=> p /\ ~q ==> r`] THEN
DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE
`k SUBSET a INSERT s /\ ~(!x. x IN k ==> x = a)
==> k DELETE a SUBSET s /\ ~(k DELETE a = {})`)) THEN
ONCE_REWRITE_TAC[GSYM UNIONS_DELETE_EMPTY] THEN
SPEC_TAC(`(k DELETE {}):(A->bool)->bool`,`v:(A->bool)->bool`) THEN
GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [SUBSET] THEN
SIMP_TAC[IN_ELIM_THM; UNIONS_SUBSET] THEN STRIP_TAC THEN
FIRST_X_ASSUM(X_CHOOSE_TAC `s:A->bool` o
REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN
MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `u DIFF s:A->bool` THEN
ASM_SIMP_TAC[] THEN ASM SET_TAC[]]);;
let TOPSPACE_COFINITE_TOPOLOGY = prove
(`!u:A->bool. topspace(cofinite_topology u) = u`,
GEN_TAC THEN REWRITE_TAC[topspace; OPEN_IN_COFINITE_TOPOLOGY] THEN
MATCH_MP_TAC(SET_RULE
`(!s. P s ==> s SUBSET u) /\ P u ==> UNIONS {s | P s} = u`) THEN
REWRITE_TAC[DIFF_EQ_EMPTY; FINITE_EMPTY] THEN SET_TAC[]);;
let CLOSED_IN_COFINITE_TOPOLOGY = prove
(`!u s:A->bool.
closed_in (cofinite_topology u) s <=>
s = u \/ s SUBSET u /\ FINITE s`,
REWRITE_TAC[closed_in; OPEN_IN_COFINITE_TOPOLOGY;
TOPSPACE_COFINITE_TOPOLOGY] THEN
REPEAT GEN_TAC THEN
ASM_CASES_TAC `(s:A->bool) SUBSET u` THEN
ASM_SIMP_TAC[SET_RULE `s SUBSET u ==> u DIFF (u DIFF s) = s`] THEN
ASM SET_TAC[]);;
let T1_SPACE_COFINITE_TOPOLOGY = prove
(`!u:A->bool. t1_space(cofinite_topology u)`,
REWRITE_TAC[T1_SPACE_CLOSED_IN_SING; CLOSED_IN_COFINITE_TOPOLOGY] THEN
SIMP_TAC[FINITE_SING; SING_SUBSET; TOPSPACE_COFINITE_TOPOLOGY]);;
let T1_SPACE_COFINITE_TOPOLOGY_MINIMAL = prove
(`!top s:A->bool.
t1_space top /\ closed_in (cofinite_topology(topspace top)) s
==> closed_in top s`,
REWRITE_TAC[CLOSED_IN_COFINITE_TOPOLOGY] THEN
MESON_TAC[T1_SPACE_CLOSED_IN_FINITE; CLOSED_IN_TOPSPACE]);;
let COFINITE_EQ_DISCRETE_TOPOLOGY = prove
(`!u:A->bool. FINITE u ==> cofinite_topology u = discrete_topology u`,
SIMP_TAC[TOPOLOGY_EQ; OPEN_IN_DISCRETE_TOPOLOGY; OPEN_IN_COFINITE_TOPOLOGY;
FINITE_DIFF] THEN
SET_TAC[]);;
let CONNECTED_IN_COFINITE_TOPOLOGY = prove
(`!u s:A->bool.
INFINITE s /\ s SUBSET u ==> connected_in (cofinite_topology u) s`,
REWRITE_TAC[CONNECTED_IN_CLOSED_IN; CLOSED_IN_COFINITE_TOPOLOGY] THEN
SIMP_TAC[TOPSPACE_COFINITE_TOPOLOGY] THEN REPEAT STRIP_TAC THENL
[ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[];
ASM_MESON_TAC[INFINITE; FINITE_SUBSET; FINITE_UNION]]);;
let CONNECTED_SPACE_COFINITE_TOPOLOGY = prove
(`!u:A->bool.
connected_space(cofinite_topology u) <=>
(?a. u SUBSET {a}) \/ INFINITE u`,
GEN_TAC THEN ASM_CASES_TAC `FINITE(u:A->bool)` THEN
ASM_SIMP_TAC[INFINITE; COFINITE_EQ_DISCRETE_TOPOLOGY] THEN
REWRITE_TAC[CONNECTED_SPACE_DISCRETE_TOPOLOGY] THEN
REWRITE_TAC[GSYM CONNECTED_IN_TOPSPACE] THEN
MATCH_MP_TAC CONNECTED_IN_COFINITE_TOPOLOGY THEN
ASM_REWRITE_TAC[INFINITE; TOPSPACE_COFINITE_TOPOLOGY; SUBSET_REFL]);;
let COMPACT_IN_COFINITE_TOPOLOGY = prove
(`!u s:A->bool. compact_in(cofinite_topology u) s <=> s SUBSET u`,
REPEAT GEN_TAC THEN
REWRITE_TAC[COMPACT_IN_FIP; CLOSED_IN_COFINITE_TOPOLOGY] THEN
REWRITE_TAC[TOPSPACE_COFINITE_TOPOLOGY] THEN
ASM_CASES_TAC `(s:A->bool) SUBSET u` THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `U:(A->bool)->bool` THEN STRIP_TAC THEN
ASM_CASES_TAC `FINITE(U:(A->bool)->bool)` THEN
ASM_SIMP_TAC[SUBSET_REFL] THEN
SUBGOAL_THEN `?k:A->bool. k IN U /\ k SUBSET u /\ FINITE k`
STRIP_ASSUME_TAC THENL
[SUBGOAL_THEN `~((U:(A->bool)->bool) SUBSET {u})` MP_TAC THENL
[ASM_MESON_TAC[FINITE_SUBSET; FINITE_SING];
REWRITE_TAC[SUBSET; IN_SING] THEN ASM SET_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN `?V. V SUBSET U /\ FINITE V /\ INTERS U :A->bool = INTERS V`
MP_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
SUBGOAL_THEN `U = k INSERT (U DELETE (k:A->bool))` SUBST1_TAC THENL
[ASM SET_TAC[]; REWRITE_TAC[INTER_INTERS; INTERS_INSERT]] THEN
ASM_SIMP_TAC[SET_RULE `a IN s ==> (s DELETE a = {} <=> s = {a})`] THEN
COND_CASES_TAC THENL [ASM_MESON_TAC[FINITE_SING]; ALL_TAC] THEN
MATCH_MP_TAC(MESON[] `!k. (?s. P(k INSERT k INSERT s)) ==> (?s. P s)`) THEN
EXISTS_TAC `k:A->bool` THEN ONCE_REWRITE_TAC[INTERS_INSERT] THEN
REWRITE_TAC[INTER_INTERS; FINITE_INSERT; NOT_INSERT_EMPTY] THEN
SUBGOAL_THEN `FINITE {k INTER t:A->bool | t | t IN U DELETE k}` MP_TAC THENL
[MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `{l:A->bool | l SUBSET k}` THEN
ASM_SIMP_TAC[FINITE_POWERSET] THEN
GEN_REWRITE_TAC I [SUBSET] THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN
REWRITE_TAC[IN_ELIM_THM; INTER_SUBSET];
REWRITE_TAC[SIMPLE_IMAGE] THEN
GEN_REWRITE_TAC LAND_CONV [FINITE_IMAGE_EQ] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `V:(A->bool)->bool` THEN
ASM_CASES_TAC `V:(A->bool)->bool = {}` THEN
ASM_REWRITE_TAC[IMAGE_CLAUSES; IMAGE_EQ_EMPTY] THENL
[ASM_MESON_TAC[FINITE_EMPTY; FINITE_DELETE]; STRIP_TAC] THEN
CONJ_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN
REWRITE_TAC[INTER_IDEMPOT; INTER_INTERS; INTERS_INSERT] THEN
ASM_REWRITE_TAC[IMAGE_EQ_EMPTY; SET_RULE
`{g y | y IN IMAGE f s} = {g(f x) | x IN s}`] THEN
REWRITE_TAC[SIMPLE_IMAGE; SET_RULE `k INTER k INTER s = k INTER s`]]);;
let COMPACT_SPACE_COFINITE_TOPOLOGY = prove
(`!u:A->bool. compact_space(cofinite_topology u)`,
REWRITE_TAC[compact_space; COMPACT_IN_COFINITE_TOPOLOGY] THEN
REWRITE_TAC[TOPSPACE_COFINITE_TOPOLOGY; SUBSET_REFL]);;
let HAUSDORFF_SPACE_SUBTOPOLOGY_COFINITE_TOPOLOGY = prove
(`!u s:A->bool.
hausdorff_space (subtopology (cofinite_topology u) s) <=>
FINITE(u INTER s)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[SUBTOPOLOGY_RESTRICT] THEN
REWRITE_TAC[TOPSPACE_COFINITE_TOPOLOGY] THEN
MP_TAC(SET_RULE `u INTER (s:A->bool) SUBSET u`) THEN
SPEC_TAC(`u INTER (s:A->bool)`,`s:A->bool`) THEN
REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THENL
[ASM_CASES_TAC `?a:A. s SUBSET {a}` THENL
[ASM_MESON_TAC[FINITE_SUBSET; FINITE_SING]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
`~(?a. s SUBSET {a}) ==> ?a b. a IN s /\ b IN s /\ ~(a = b)`)) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `b:A`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`a:A`; `b:A`] o
GEN_REWRITE_RULE I [hausdorff_space]) THEN
ASM_SIMP_TAC[TOPSPACE_SUBTOPOLOGY_SUBSET; TOPSPACE_COFINITE_TOPOLOGY] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
REWRITE_TAC[OPEN_IN_SUBTOPOLOGY_ALT; FORALL_IN_GSPEC] THEN
REWRITE_TAC[IN_INTER; OPEN_IN_COFINITE_TOPOLOGY] THEN
X_GEN_TAC `t:A->bool` THEN STRIP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
X_GEN_TAC `t':A->bool` THEN STRIP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `(u DIFF t) UNION (u DIFF t'):A->bool` THEN
ASM_REWRITE_TAC[FINITE_UNION] THEN ASM SET_TAC[];
ASM_SIMP_TAC[hausdorff_space; TOPSPACE_SUBTOPOLOGY_SUBSET;
TOPSPACE_COFINITE_TOPOLOGY] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN STRIP_TAC THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; OPEN_IN_SUBTOPOLOGY_ALT] THEN
REWRITE_TAC[EXISTS_IN_GSPEC; OPEN_IN_COFINITE_TOPOLOGY] THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
EXISTS_TAC `(x:A) INSERT (u DIFF s)` THEN
EXISTS_TAC `(y:A) INSERT (u DIFF s)` THEN
REWRITE_TAC[NOT_INSERT_EMPTY] THEN
REPEAT CONJ_TAC THEN
TRY(MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `s:A->bool`) THEN
ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);;
let HAUSDORFF_SPACE_COFINITE_TOPOLOGY = prove
(`!u:A->bool. hausdorff_space(cofinite_topology u) <=> FINITE u`,
GEN_TAC THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_TOPSPACE] THEN
REWRITE_TAC[HAUSDORFF_SPACE_SUBTOPOLOGY_COFINITE_TOPOLOGY] THEN
REWRITE_TAC[TOPSPACE_COFINITE_TOPOLOGY; INTER_IDEMPOT]);;
(* ------------------------------------------------------------------------- *)
(* Countable-complement (cocountable) topology. *)
(* ------------------------------------------------------------------------- *)
let cocountable_topology = new_definition
`cocountable_topology (u:A->bool) =
topology ({} INSERT {s | s SUBSET u /\ COUNTABLE(u DIFF s)})`;;
let OPEN_IN_COCOUNTABLE_TOPOLOGY = prove
(`!u s:A->bool.
open_in (cocountable_topology u) s <=>
s = {} \/ s SUBSET u /\ COUNTABLE(u DIFF s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[cocountable_topology] THEN
W(MP_TAC o fst o EQ_IMP_RULE o PART_MATCH (lhand o rand)
(CONJUNCT2 topology_tybij) o rator o lhand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[istopology];
DISCH_THEN SUBST1_TAC THEN
GEN_REWRITE_TAC LAND_CONV [GSYM IN] THEN
REWRITE_TAC[IN_ELIM_THM; IN_INSERT]] THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_INSERT] THEN
REWRITE_TAC[IN_INSERT; INTER_EMPTY] THEN CONJ_TAC THENL
[REWRITE_TAC[IN_ELIM_THM] THEN
X_GEN_TAC `s:A->bool` THEN STRIP_TAC THEN
X_GEN_TAC `t:A->bool` THEN STRIP_TAC THEN DISJ2_TAC THEN
REWRITE_TAC[SET_RULE `u DIFF s INTER t = (u DIFF s) UNION (u DIFF t)`] THEN
ASM_SIMP_TAC[COUNTABLE_UNION] THEN ASM SET_TAC[];
X_GEN_TAC `k:(A->bool)->bool` THEN REWRITE_TAC[EMPTY_UNIONS] THEN
REWRITE_TAC[TAUT `p ==> q \/ r <=> p /\ ~q ==> r`] THEN
DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE
`k SUBSET a INSERT s /\ ~(!x. x IN k ==> x = a)
==> k DELETE a SUBSET s /\ ~(k DELETE a = {})`)) THEN
ONCE_REWRITE_TAC[GSYM UNIONS_DELETE_EMPTY] THEN
SPEC_TAC(`(k DELETE {}):(A->bool)->bool`,`v:(A->bool)->bool`) THEN
GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [SUBSET] THEN
SIMP_TAC[IN_ELIM_THM; UNIONS_SUBSET] THEN STRIP_TAC THEN
FIRST_X_ASSUM(X_CHOOSE_TAC `s:A->bool` o
REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN
MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `u DIFF s:A->bool` THEN
ASM_SIMP_TAC[] THEN ASM SET_TAC[]]);;
let TOPSPACE_COCOUNTABLE_TOPOLOGY = prove
(`!u:A->bool. topspace(cocountable_topology u) = u`,
GEN_TAC THEN REWRITE_TAC[topspace; OPEN_IN_COCOUNTABLE_TOPOLOGY] THEN
MATCH_MP_TAC(SET_RULE
`(!s. P s ==> s SUBSET u) /\ P u ==> UNIONS {s | P s} = u`) THEN
REWRITE_TAC[DIFF_EQ_EMPTY; COUNTABLE_EMPTY] THEN SET_TAC[]);;
let CLOSED_IN_COCOUNTABLE_TOPOLOGY = prove
(`!u s:A->bool.
closed_in (cocountable_topology u) s <=>
s = u \/ s SUBSET u /\ COUNTABLE s`,
REWRITE_TAC[closed_in; OPEN_IN_COCOUNTABLE_TOPOLOGY;
TOPSPACE_COCOUNTABLE_TOPOLOGY] THEN
REPEAT GEN_TAC THEN
ASM_CASES_TAC `(s:A->bool) SUBSET u` THEN
ASM_SIMP_TAC[SET_RULE `s SUBSET u ==> u DIFF (u DIFF s) = s`] THEN
ASM SET_TAC[]);;
let T1_SPACE_COCOUNTABLE_TOPOLOGY = prove
(`!u:A->bool. t1_space(cocountable_topology u)`,
REWRITE_TAC[T1_SPACE_CLOSED_IN_SING; CLOSED_IN_COCOUNTABLE_TOPOLOGY] THEN
SIMP_TAC[COUNTABLE_SING; SING_SUBSET; TOPSPACE_COCOUNTABLE_TOPOLOGY]);;
let COCOUNTABLE_EQ_DISCRETE_TOPOLOGY = prove
(`!u:A->bool. COUNTABLE u ==> cocountable_topology u = discrete_topology u`,
SIMP_TAC[TOPOLOGY_EQ; OPEN_IN_DISCRETE_TOPOLOGY; OPEN_IN_COCOUNTABLE_TOPOLOGY;
COUNTABLE_DIFF] THEN
SET_TAC[]);;
let CONNECTED_IN_COCOUNTABLE_TOPOLOGY = prove
(`!u s:A->bool.
~COUNTABLE s /\ s SUBSET u
==> connected_in (cocountable_topology u) s`,
REWRITE_TAC[CONNECTED_IN_CLOSED_IN; CLOSED_IN_COCOUNTABLE_TOPOLOGY] THEN
SIMP_TAC[TOPSPACE_COCOUNTABLE_TOPOLOGY] THEN REPEAT STRIP_TAC THENL
[ASM SET_TAC[]; ASM SET_TAC[]; ASM SET_TAC[];
ASM_MESON_TAC[COUNTABLE_SUBSET; COUNTABLE_UNION]]);;
let CONNECTED_SPACE_COCOUNTABLE_TOPOLOGY = prove
(`!u:A->bool.
connected_space(cocountable_topology u) <=>
(?a. u SUBSET {a}) \/ ~COUNTABLE u`,
GEN_TAC THEN ASM_CASES_TAC `COUNTABLE(u:A->bool)` THEN
ASM_SIMP_TAC[COCOUNTABLE_EQ_DISCRETE_TOPOLOGY] THEN
REWRITE_TAC[CONNECTED_SPACE_DISCRETE_TOPOLOGY] THEN
REWRITE_TAC[GSYM CONNECTED_IN_TOPSPACE] THEN
MATCH_MP_TAC CONNECTED_IN_COCOUNTABLE_TOPOLOGY THEN
ASM_REWRITE_TAC[TOPSPACE_COCOUNTABLE_TOPOLOGY; SUBSET_REFL]);;
let COMPACT_IN_COCOUNTABLE_TOPOLOGY = prove
(`!u s:A->bool. compact_in(cocountable_topology u) s <=>
s SUBSET u /\ FINITE s`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `(s:A->bool) SUBSET u` THENL
[EQ_TAC THEN
ASM_SIMP_TAC[FINITE_IMP_COMPACT_IN; TOPSPACE_COCOUNTABLE_TOPOLOGY];
ASM_MESON_TAC[COMPACT_IN_SUBSET_TOPSPACE;
TOPSPACE_COCOUNTABLE_TOPOLOGY]] THEN
GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[FINITE_CARD_LT; CARD_NOT_LT; CARD_LE_EQ_SUBSET] THEN
DISCH_THEN(X_CHOOSE_THEN `t:A->bool` STRIP_ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP CARD_FINITE_CONG) THEN
REWRITE_TAC[REWRITE_RULE[INFINITE] num_INFINITE] THEN
ASM_CASES_TAC `t:A->bool = {}` THEN ASM_REWRITE_TAC[FINITE_EMPTY] THEN
DISCH_TAC THEN
REWRITE_TAC[COMPACT_IN_FIP; CLOSED_IN_COCOUNTABLE_TOPOLOGY] THEN
ASM_REWRITE_TAC[TOPSPACE_COCOUNTABLE_TOPOLOGY] THEN
DISCH_THEN(MP_TAC o SPEC `{t DIFF {x:A} | x | x IN t}`) THEN
REWRITE_TAC[NOT_IMP; FORALL_IN_GSPEC] THEN REPEAT CONJ_TAC THENL
[X_GEN_TAC `a:A` THEN DISCH_TAC THEN DISJ2_TAC THEN
CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `t:A->bool` THEN
REWRITE_TAC[SUBSET_DIFF; COUNTABLE; ge_c] THEN
ASM_MESON_TAC[CARD_EQ_SYM; CARD_EQ_IMP_LE];
ALL_TAC;
REWRITE_TAC[INTERS_GSPEC] THEN ASM SET_TAC[]] THEN
REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE; SIMPLE_IMAGE] THEN
X_GEN_TAC `v:A->bool` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`t SUBSET s ==> ~(t INTER u = {}) ==> ~(s INTER u = {})`)) THEN
REWRITE_TAC[SET_RULE
`IMAGE (\x. t DIFF {x}) v = {t DIFF u | u IN IMAGE (\x. {x}) v}`] THEN
REWRITE_TAC[GSYM DIFF_UNIONS] THEN
REWRITE_TAC[GSYM SIMPLE_IMAGE; UNIONS_SINGS] THEN
ASM_SIMP_TAC[SET_RULE `v SUBSET t ==> (t DIFF v = {} <=> t = v)`] THEN
ASM_MESON_TAC[]);;
let COMPACT_SPACE_COCOUNTABLE_TOPOLOGY = prove
(`!u:A->bool. compact_space(cocountable_topology u) <=> FINITE u`,
REWRITE_TAC[compact_space; COMPACT_IN_COCOUNTABLE_TOPOLOGY] THEN
REWRITE_TAC[TOPSPACE_COCOUNTABLE_TOPOLOGY; SUBSET_REFL]);;
let HAUSDORFF_SPACE_SUBTOPOLOGY_COCOUNTABLE_TOPOLOGY = prove
(`!u s:A->bool.
hausdorff_space (subtopology (cocountable_topology u) s) <=>
COUNTABLE(u INTER s)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[SUBTOPOLOGY_RESTRICT] THEN
REWRITE_TAC[TOPSPACE_COCOUNTABLE_TOPOLOGY] THEN
MP_TAC(SET_RULE `u INTER (s:A->bool) SUBSET u`) THEN
SPEC_TAC(`u INTER (s:A->bool)`,`s:A->bool`) THEN
REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THENL
[ASM_CASES_TAC `?a:A. s SUBSET {a}` THENL
[ASM_MESON_TAC[COUNTABLE_SUBSET; COUNTABLE_SING]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
`~(?a. s SUBSET {a}) ==> ?a b. a IN s /\ b IN s /\ ~(a = b)`)) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `b:A`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`a:A`; `b:A`] o
GEN_REWRITE_RULE I [hausdorff_space]) THEN
ASM_SIMP_TAC[TOPSPACE_SUBTOPOLOGY_SUBSET;
TOPSPACE_COCOUNTABLE_TOPOLOGY] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
REWRITE_TAC[OPEN_IN_SUBTOPOLOGY_ALT; FORALL_IN_GSPEC] THEN
REWRITE_TAC[IN_INTER; OPEN_IN_COCOUNTABLE_TOPOLOGY] THEN
X_GEN_TAC `t:A->bool` THEN STRIP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
X_GEN_TAC `t':A->bool` THEN STRIP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC COUNTABLE_SUBSET THEN
EXISTS_TAC `(u DIFF t) UNION (u DIFF t'):A->bool` THEN
ASM_REWRITE_TAC[COUNTABLE_UNION] THEN ASM SET_TAC[];
ASM_SIMP_TAC[hausdorff_space; TOPSPACE_SUBTOPOLOGY_SUBSET;
TOPSPACE_COCOUNTABLE_TOPOLOGY] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN STRIP_TAC THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; OPEN_IN_SUBTOPOLOGY_ALT] THEN
REWRITE_TAC[EXISTS_IN_GSPEC; OPEN_IN_COCOUNTABLE_TOPOLOGY] THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
EXISTS_TAC `(x:A) INSERT (u DIFF s)` THEN
EXISTS_TAC `(y:A) INSERT (u DIFF s)` THEN
REWRITE_TAC[NOT_INSERT_EMPTY] THEN
REPEAT CONJ_TAC THEN
TRY(MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC `s:A->bool`) THEN
ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);;
let HAUSDORFF_SPACE_COCOUNTABLE_TOPOLOGY = prove
(`!u:A->bool. hausdorff_space(cocountable_topology u) <=> COUNTABLE u`,
GEN_TAC THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_TOPSPACE] THEN
REWRITE_TAC[HAUSDORFF_SPACE_SUBTOPOLOGY_COCOUNTABLE_TOPOLOGY] THEN
REWRITE_TAC[TOPSPACE_COCOUNTABLE_TOPOLOGY; INTER_IDEMPOT]);;
let KC_SPACE_COCOUNTABLE_TOPOLOGY = prove
(`!u:A->bool. kc_space(cocountable_topology u)`,
REWRITE_TAC[kc_space; CLOSED_IN_COCOUNTABLE_TOPOLOGY;
COMPACT_IN_COCOUNTABLE_TOPOLOGY] THEN
SIMP_TAC[FINITE_IMP_COUNTABLE]);;
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